Mathematical Completion
by Seien24
Summary: The Cybertronian Golden Age was a time of peace and creativity. Starscream, then a student at the Academy of Science and Technology, discovers a new proof for himself. Slash, Young!Starscream/Skyfire, mathkink.


The Academy of Science and Technology was, for Starscream, the most exciting place in all of Cybertron. Two of the planet's three moons rose one after the other over the otherwise empty inner courtyard where the young jet was kneeling on the ground, surrounded by datapads: one covered in scribbled calculations, a glossary, a mathematical dictionary, a compendium of axioms. The Academy and its tutors guided their students on a voyage of discovery - a thesis was written when the student felt ready to finish up and leave, but other than that what guided the young scholars was fascination, enthusiasm, determination and curiosity. Many subjects were explored at the Academy - logic, physics, chemistry, every kind of biology under the stars, geology, xenology, geography - but Starscream's passion and joy was to be found in the complex and enthralling syntax of pure mathematics. The perfect universal truth expressed within its depths attracted Starscream utterly. The jet seemed infused with a passionate and dramatic determination to _understand_; he devoured information and demanded more, driven to get on top of his chosen field and then some. He would sulk for days when a problem confounded him and shriek with joy when he solved it. Sometimes he would be quite content to puzzle over the same problem for deca-cycles and refuse to allow the tutors to explain it to him or, in some cases, even see it at all. Finally, he would storm into his tutors' offices and hand them datapads filled with reams of calculations and finishing up with "Q.E.D." - and soaking up the praise and adulation like energon.

He worked single-mindedly, scrawling symbols and notations and abbreviated notes, stylus flying across the surface of the datapad. He was completely alone in the courtyard, and so there was nobody to hear the ear-splitting screech of triumph as he made the final connection of logic and suddenly the whole problem unfolded before him, making perfect and elegant _sense_. He clutched the datapad containing his proof with both hands, scrambling to his feet and turning to leave the courtyard. He'd not gone four steps when he turned, backtracked and gathered his other datapads. He ran from the courtyard, into the elevator and crashed into his quarters.

Skyfire jumped, surprised, as his roommate barreled through the door. In contrast to Starscream's nigh-hyperactive exuberance, the white spacecraft was almost sedate in his calmness and composure. He smiled at the smaller jet as Starscream rearranged the datapads on his desk fastidiously, optics bright with happiness.

"A success, then?" Skyfire intoned, a little amused at Starscream's predictable excitedness.

The dainty jet finished putting his things away and turned to Skyfire, his ruby optics fixing themselves on his friend's face. Skyfire shivered suddenly in anticipation - there was no mistaking the look in those blazing optics. Starscream had decided to share.

Quite beyond the thrill of a successful solution, Starscream adored the _process _of discovering a proof. He loved to slowly pull away the covers that obscured such elegant, beautiful truths using logic and reason, leaving the facts and verities laid out naked before him. An upside to his existence was that his beloved companion shared his passion in this respect.

"Did you know," began Starscream, his tone nonchalant. "that if you put _n_ equally spaced marks around the circumference of a unit circle, draw chords from any one mark that connect it to all the others, and multiply the lengths together, that the product will always be _n_?"

The comment was an insouciant opening line, as offhand as "accidentally" brushing a hand across a wing. Skyfire's optics darkened slightly.

"Is that so?" He murmured.

Starscream walked over, leaning on his companion's desk. He raked his optics almost tangibly over Skyfire's armour, mischeivousness mingled with desire and barely concealed at all in his expression.

"Well, we must consider this unit circle to be the complex unit circle. As you know, in complex number analysis, the roots of unity are _n_ points equally spaced around a unit circle, starting at the complex number 1 - collectively, the _n _marks are the _n_th roots of 1." Starscream purred.

He reached up and cupped Skyfire's face in his hand, stroking over the planes and angles of the spacecraft's dermaplating and helm. Skyfire's optics guttered and his glossa flicked out over his lower lip.

"Continue," he murmured heatedly, his own hand settling firmly on Starscream's canopy. His engines were humming softly now, indicative of his arousal. To the two scientifically-programmed Transformers, the mathematics had of themselves a natural beauty, a flawless allure. They were as exquisite as any attractive mech, perhaps even more so because quite apart from the shared qualities of streamlined grace, poise and elegance, numbers were theoretical - truly perfect, all-encompassing in their consummate magnificence.

"Let us call one of the other endpoints _r_, then the segment connecting it to the number 1 is the vector that represents the complex number 1-_r_; which means the length of that segment is the absolute value of 1-_r._"

Starscream pressed against Skyfire's hand, his voice mellowed into something light and airy with lust. His hand slipped to Skyfire's neck, stroking the tension cables there gently.

"So, with that in mind, note the polynomial identity _z__n__-1 (z-r__1__)(z-r__2__)(z-r__3__)…(z-r__n_which holds over the complex numbers."

Skyfire moaned.

Starscream's lips quirked prettily into a smile as he continued his onslaught of elegance, his hand now resting affectionately on Skyfire's shoulder.

"Now, let us make the following substitutions."

Skyfire's optics brightened. Starscream straddled one of Skyfire's massive thighs, leaning against glossy white armour and letting the vibrations from his turbines exchange and intermingle with those from the spacecraft's engines. He mewled softly, fingers on the other's chest as his lover rubbed his wings, before continuing.

"The first root of 1 is clearly just 1, so _r__1 _becomes 1. Then, also, replace all instances of _z _with _z_ 1. So, _(Z 1)__n __-1 ((Z 1)-1)((Z 1)-r__1__)((Z 1)-r__2__)((Z 1)-r__3__)…((Z 1)-r__n_We can then get a second line by the binomial expansion on the left, and the obvious simplification on the right: this gives _Z__n__ nZ__n-1__ … nZ 1 -1 Z(Z 1-r__2__)…(Z 1-r__n_"

Skyfire gasped softy, lips parted, the numbers flooding his mind and wrapping him in a haze of desire. His trembling hands gripped Starscream's back and wings and any other part of him they touched. The clean refinement and grace of the numbers affected him on a deep, completely natural level. Starscream's voice washed over him, sending him into raptures of ecstasy with the sheer beauty of the ideas, the exquisite simplicity.

"Nnh... Starscream, more..."

Starscream pressed his forehead against Skyfire's shoulder briefly before looking up and staring into those azure optics, his voice rough and needy.

"Since the 1s cancel on the left, we can divide both sides by _z _to get _Z__n-1__ nZ__n-2__ … n (Z 1-r__2__)(Z 1-r__3__)…(Z 1-r__n__)." _

Skyfire shook beneath him, so very close. Starscream dragged his fingers over Skyfire's canopy as he gasped out the final part of the proof, the wonderful statement of mathematical completion that would build on and justify everything he had so far revealed, remove that final layer, top off the conjecture. His voice dropped to a lustful purr, ruby optics blazing into sapphire ones.

"And if you take the absolute value of both sides, you get simply that the lengths of all the chords have a product of _n_."

Skyfire threw his head back, crushing Starscream to him, and overloaded with a rapturous groan. His engines screamed and the heavy vibrations tore through Starscream's chassis, sent him spiralling into his own overload, clutching desperately to Skyfire, slate-grey dermaplating pressed against snow-white armour. He went limp, sprawled across his lover lazily, lapis fingers petting Skyfire's helmet. The big mech composed himself and stood, carrying Starscream, and moved over to the recharge berth dazedly. Starscream curled contentedly against his friend, nuzzling into him happily and offlining.

Starscream's existence at the Academy was a civilised and pleasurable one indeed.


End file.
